utexasedu R P RESTON FEE Humanities Social Sciences California Institute of Technology Pasadena CA 91125 mcafeehsscaltechedu In both economic and military situations agents may try to mislead rivals about their true types or plans whatever they may ID: 88048
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attacksothatthedefendercannotbecertainastowhichofthetwobattlesisthedecisiveone.Asaresult,hewillsplithisforcesbetweenthetwobattles,sendingrelativelymoreforcestothebattlethathethinksismorelikelytobedecisivebasedontheinformationgeneratedbytheattackerÕsdivisionofforces.Theattacker,knowinghowthedefenderwillrespond,hasanincentivetomanipulatethedefenderÕsbeliefsbyallocatingsomeofhisforcestothelessimportantbattle,andhopefullysendasignalthatwilldrawmostofthedefenderÕsforcesawayfromthemoreimportantbattle.Aninterestingspecialcasestudiedinapreviousversionofthispaperassumesthatiseither0or1(i.e.,onlyonebattlematters).Inthismodel,theattackerallocatesforcestoeachlocationsoastomakethedefenderuncertainastowhichlocationisgoingtobeattacked.ThediversionaryforcethreatensbutdoesnotactuallyÞghtsothecostofdivertingforcesawayfromthebattleisanopportunitycost.Thepayoffstotheattackerandthedefenderaremeasuredbythedifferenceinthesizeoftheforcesatthelocationoftheattack.TheattackerÕspayoffsarealsoallowedtovaryacrosslocationstoreßectdifferencesinthelocations.Forexample,CalaiswasaneasiertargettoattackthanNormandy.Noneofthesemodelingdetailshadanyqualitativeeffectonequilibriumbehavior.Theattackerrandomizesoverwhichlocationtoattackandthen,dependinguponthesignalingtechnology,divideshisforcesusingthestrategiesdescribedinPropositions1and2.Ourresultsdistinguishbetweentwokindsofsituations.Whenthesignalingtechnologyisnoisy,thedefenderÞndsitdifÞculttoidentifywhichlocationisbeingattackedbythelargerforceeventhoughthediversionaryforceissmall.Hence,theattackercanaffordtoattackhispreferredlocationwithmostofhisforces.ThesignalisnotveryinformativesothedefenderÕsstrategyisbasicallytodividehisforcesequallybetweenthetwolocationsandmarginallyincreasetheforcesallocatedtothelocationsignaled.TheAlliedinvasionofNormandyservesasanexampleofthiskindofsituation.Whenthedefendercaneasilydetectwhethertheattackforcetargetedatalocationissmallorlarge,theattackersometimesattackshispreferredlocationwithallofhisforcesandsometimesheusesthemainforceasadiversionandattackswithasmallforce.Theuncertaintycreatedbythisstrategymakesthesignallessinformative.Thedefendermayknowwhichlocationistargetedbythelargerforce,buthecannotbecertainastowhichbattleisgoingtobethedecisiveone.Despitethisuncertainty,herespondsbyallocatingsubstantiallymoreforcestothelocationthreatenedbytheperceivedlargerforce.Theresponsemeansthattheexpectedpayoffto JournalofEconomics&ManagementStrategyBecausetheequilibriumpayofftothesenderforrevealingtechnologyis1,thehypothesisisestablished.AmoreformalproofisgivenintheAppendix.AlthoughthesenderÕspayoffforanoisytechnologyisclearlylargerthanhispayofffortheidentitytechnology,wehavenotbeenabletoshowthatthevalueofstealthisalwayspositivewithoutrestrictingthepreferencesofthereceiver.Ifthereceiverisriskneutral,thenthereceiverÕsequilibriumstrategyisalwaystofullyinvestinthealternativesignaled(i.e.,1).Asaresult,thesenderÕsequilibriumpayoffisindependentofthereceiverÕspreferences.ThisallowsonetocomparethesenderÕsproÞtsfordifferentnoisytechnologies.Proposition5:Supposepandparenoisytechnologiesandthereceiverisriskneutral.Thenthesenderprefersptopifpislessrevealingthanp5.ApplicationsWehavepresentedthemodelinagenericformbecausemanyofthespeciÞcdetailsonpayoffsthatvarywithapplicationsarenotessentialtotheanalysis.FeintingisausefulstrategyingameswherethesecondmoverwantstodirectitsresourcesattheÞrstmoverÕschoiceoflocationinproduct/geographicalspace.TheÞrstmoverhasanincentivetodeceivethesecondmoveraboutitslocationchoicesothatthesecondmoverÕsresponsewillbemoremuted(ordelayed)thanitwouldhavebeenotherwise.Inthissection,wediscusstwoclassesof5.1MilitaryGamesThemostobviousanddirectapplicationofthemodelistomilitaryconßicts.Herethesenderistheattacker,thereceiveristhedefender,andthealternativesarelocations.TheattackerÕspayoffisaweightedaverageoftheoutcomesofthebattlesateachlocation,asmeasuredbythedifferentialinthesizeoftheforces.TheattackerÕstype,,ishisbeliefthatthebattleatlocationislikelytobedecisivetowinningthewar.Theattackerwouldliketocommitallofhisforcestothemoredecisivebattlebut,inpreparingandcarryingouthisplanofattack,theattackercannotavoidgeneratinginformationabouttherelativesizeoftheforcesbeingsenttoeachlocation.Alargerforcerequiresmoresupport,moreextensivesupplylines,andcannotmoveasquietlyorasquicklyasasmallforce.Iftheattackeriscertaintoengageinanall-outattack,itwillbemetbyanall-outdefense.Theattackermustpursueaplanof yieldsthelimitvalues4andSubstitutingthesevaluesintothesenderÕsproÞtfunctiongivesequilibriumpayoffs 2 q1 2 ifq�1 21 2 1 2q ifq1 Thus,thesenderisbetteroffwithaverynoisytechnologythanaveryrevealingtechnology.Theintuitionisthatthesenderhastoexpendatleasthalfofhisresourcesgeneratingthefalsesignalwhenthesignalingtechnologyisveryrevealingwhereasitisfreewhenthesignalingtechnologyisverynoisy.NotethatthesenderÕspayoffiscontinuousandequaltozerofor Thedividinglinebetweenthetwoclassesofsignalingtechnologiesistheidentitytechnology,.Inthiscase,thereisacontinuumofequilibriumallocations,allofwhichyieldanexpectedpayoffofzerotothesender.AsecondapproachtothequestionofwhetherthesenderprefersanoisiertechnologyistoÞxthesignalingtechnologyandconsideranincreaseinthenoise.Thefollowingpropositionestablishesthatstealthdoesnotpayifthetechnologyisrevealing.Proposition4:Supposepandparerevealingtechnologies.ThenthesenderprefersptopifpismorerevealingthanpTheintuitionisthatamorerevealingtechnologymakesitiseasiertogeneratethefalsesignalsothesendercanreducetheamountinvestedinthefeint.TheproofiseasilyseenusingFigure2.Letdenotethelowerboundoftheconvexhullofthegraphofandletthelowerendpointofthelinearsegmentson1,2.Theslopeofthechordconnecting())to(1,1)is ismoreinformativethan,theslopeofthechordonlargerthantheslopeofthelinearsegmenton.From(4),thisimpliesthattheamountinvestedinalternativeconditionalonsignal,denoted,issmallerwhenthetechnologyisthanwhenthetechnologyis7.Thevalueofmustsatisfy(5).Forexample,ifisuniformlydistributed,thisequationbecomes U(1/2)=1+2 32. signalwerefeintsnotpossible.Suchstrongresponsesbythereceiverinvitelargeattemptstofoolthereceiver,andequilibriumdictatesamixedstrategyinattemptstofoolthereceiverwithalargediversionofresourceswhenanattemptoccurs.TheÒsneakattackÓisapartofanaturalequilibriumthatariseswithpositiveprobabilitywhenthereceivercanreactstronglytothesignal.Thetheoryalsopresentsstrikingcomparativestaticresults.MorenoisebeneÞtsthesenderinanoisyworld,butnotinarevealingworld.Inarevealingworld,makingthetechnologylessnoisymakestheimpactofthefeintgreaterandlesscostly.Wehavetriedtokeepthemodelassimpleaspossibleinordertouncovertheessentiallogicoffeints.Themodelcanbeextendedinanumberofdirections.Forexample,thesignaldoesnothavetobebinary.Infact,wehavebeenabletoshowthat,undercertainconditions,themodelwithacontinuumofsignalsisisomorphictoamodelwithbinarysignals.TheessentialpropertiesarethoseofthepayofffunctionsthatimplythatthesenderÕsoptimalperceivedtypeisadecreasingfunctionofitstype.Asaresult,thesenderÕsbestreplytoanondecreasingstrategybythereceivermaynotbenondecreasing,andapurestrategyequilibriummaynotexist.TheprimaryassumptionthataccountsforthesimplicityoftheanalysisistheassumptionthatthesenderÕspayoffislinearinhisaction.WearecurrentlyexploringthecaseinwhichthesenderÕspayoffisstrictlyconcave.Thisappendixcontainstheproofsthatarenotinthetext.ProofofLemma1.BydeÞnitionThethirdequalityfollowsfromsymmetryofProofofProposition1.SubstitutingthesenderÕsbestreplyinto(1)and(2),itfollowsfromthesymmetryofthatö.Ifthetechnologyisnoisy,thenthereceiverÕsexpectationofgivensignalgivenby ofthereceiverÕsinvestment.WeallowthesendertorandomizeovertheallocationofresourcesgivenhistypeanddenotesuchanallocationThereceiverweightsthealternativesinthesamewayasthesender.Whenthereceiverbelievesthat,thenhispayofffromallocatingtoalternativeisgivenbywhereisstrictlyincreasing,astrictlyconcavefunction.NotethatthereceiverÕsrewardfromeachalternativedoesnotdirectlydependuponthesenderÕsinvestment.Hence,thereceiveralwayswantstoallocatemoreresourcestothealternativethathebelieveshasthehigherweight,regardlessofthesenderÕsallocationofresources.Infact,itisstraightforwardtoshowthatthereceiverÕsoptimalresponse)isstrictlyincreasinginöTheassumptionthatthereceiverÕspreferencesacrossalternativesmirrorthoseofthesendercreatesaconßictbetweenthetwoplayers.Thesenderwantsamismatch,inwhichheinvestsallofhisresourcesandthereceiverinvestsnoneofhisresourcesinthepreferredalternative.Butdoingsoinvitesaresponsefromthereceiver,whoalsowantstoinvestheavilyinthepreferredalternative,whichcouldeliminateanyrewardtothesender.ThefollowingassumptionimpliesthatthereceiverÕsbestreplytothesenderÕsstrategyofinvestingonlyinthepreferredalternativeistomatchthesenderÕsactionandinvestallofhisresourcesintothesamealternative.Assumption2: 2]U(1)+(1E[q|q1 Assumption2willproveusefulinrulingoutuninterestingbound-arycases.Ingeneral,thereceiverdoesnotknowbutmustinferitfromthesignalgeneratedbythesenderÕsallocation.HeupdateshisbeliefsusingBayesÕrule.Theposteriordensityofgivensignalisgivenby Theposteriordensityofgivensignalisgivenby 5.Notethatsymmetryofimpliesthat 2]=1E[q|q�1 2]. JournalofEconomics&ManagementStrategyOneapproachtothisquestionistostudythelimitcases.Asthesignalingtechnologygetsverynoisy,)convergesto0if if01ifInthelimit,anegligiblefeintmakesthelikelihoodofgeneratingsignalsequallylikely.Thesignalisessentiallyuninformative.Thereceiversplitshisresourcesequallybetweenthetwoalternativesandthesenderfullyinvestsinhispreferredalternative.Hence,inthelimit,theexpectedpayofftothesenderconvergesto 2ifq�1 21 2qifq1 Bycontrast,asthetechnologybecomesmorerevealing,)con-vergesto0if 2�x01 2ifx=1 1if1 Inthelimit,theamountthatthesenderneedstoinvesttogeneratethefalsesignalforcertainapproaches(butisneverequalto) .Hightypesrandomizebetweeninvestingfullyandanamountslightlylessthan inalternative;lowtypesrandomizebetweeninvestingfullyandanamountthatisslightlylessthan inalternative.Asaresult,ifthereceiverobservessignal,hedoesnotknowwhetherheisfacingahightypewhoisinvestingfullyinalternativeoralowtypewhoisfeinting.Similarly,ifheobservessignal,hedoesnotknowwhetherheisfacingahightypewhoisfeintingoralowtypewhoisinvestingfullyinalternative.Recallthat,toensurethatthesenderiswillingtorandomizeinthisway,thereceiverÕsstrategymustsatisfy(4).This JournalofEconomics&ManagementStrategytodecoyStandardOilintoblockingthewrongpipelinerouteareanexampleofthisstrategy.Manycompaniesareaccusedofproductannouncementswhosepurposeistomislead.Suchproductsareknownasvaporware.Inmostcases,thepurposeistomisleadconsumers.Forexample,Microsofthasbeenaccusedofannouncingproductstodeterconsumersfromswitchingtosuperiorproducts(seeMcAfee(2002),pp.348Ð349).TheseexamplesarenotwelldescribedbythepresenttheorybecausetheinterestsoftheconsumersalignwiththeÞrmsÕinterestsinthestateoftheworldwheretheproductexists.However,someinstancesmayinvolvemisdirectionoftheresearchofrivalcompanies.MicrosofthasrecentlydroppedtheÞlesystemknownasWinFS,nowalmost10yearsoverdue,fromthenewoperatingsystemcode-namedLonghorn.BecausetherearenosigniÞcantcompetingproductstotheWindowsXPÞlesystemNTFS,andthedelaysolongastosuggestintent,itisplausibletothinktheWinFSvaporwareisaimedatcompetitors,especiallyLinux,ratherthanconsumers.EvenwithoutthenewÞlesystem,Longhornitselfhasbeenpushedbacktwoyears,sothatsomearesuggestingitbecalledÒlongwait.ÓAttradeshows,ÞrmshaveallegedlyshowcasedmodiÞedversionsoftheirnewproductslestrivalscopycattheirdesignsbeforethemarketopens.Langinier(2000)citesanecdotalevidenceofÞrmspatentingÒdeadendsÓinordertosendcompetitorsinthewrongresearchdirectionandprovidesaninterestingexamplefromthepharmaceuticalindustryinwhichthepatentingÞrmcouldhavetriedtopursueafeintingstrategy.6.ConclusionWepresentasignalingtheoryoffeintsbasedonrationalplayers,whichemphasizescostlyactionstakentomislead.Whenmisdirectioniscostly,thesenderwillnotattempttomisleadunlessthereisapayofftofoolingthereceiver;thusfeintsmustbeatleastpartiallysuccessfultooccur.Ourtheoryshowsthattherearetwotypesoffeints,andthetypethatarisesdependsontheaccuracyornoisinessofthesignalingtechnology.Inanoisyworld,thereceiverdoesnotrespondstronglytothesignal(becausethesignalisnotveryinformative),sotypicallylargeeffortsatmisdirectionarenotworthwhile,andinvestmentinafeintismodest.Incontrast,iftheworldisnotnoisy,whichwecallrevealing,signalsaremeaningfulandthereceiverwouldrespondverystronglytothe10.SeetheforthcomingarticlebyLeanderKahney,ÒVaporwarePhantomHauntsUsAll,ÓinWiredMagazine;http://www.wired.com/news/culture/0,1284,66195,00.html,January7,2005. JournalofEconomics&ManagementStrategyastheprobabilitythatthereceiverobservessignalconditionalonthesenderallocatingunitstoalternative.WewillimposethefollowingregularityconditionsonthesignalingtechnologyAssumption1:isstrictlyincreasinganddifferentiable,(ii)1,(iii)Condition(i)statesthattheprobabilityofgeneratingsignalcreaseswiththeamountofresourcesallocatedtoalternative.Condition(ii)imposessymmetry.Itimpliesthattheprobabilityofthereceivergettingsignalwhenthesenderallocatesunitstoalternativeisthesameastheprobabilityofgettingsignalwhenthesenderallocatesunitstoalternative.Notethatthisconditionimpliesthat )isequal .Condition(iii)statesthatifthesenderallocatesnoresourcesto,thenthereceiveriscertaintogetsignal.Bysymmetry,ifthesenderallocatesallofhisresourcestoalternative,thereceiveriscertaintogetsignal.Inotherwords,thesenderhastoallocateresourcestobothalternativesifhewantsthereceivertobeuncertainaboutwhichalternativeisgettingmoreresources.Inresponsetothesignal,thereceiverdivideshisresourcesbetweenthetwoalternatives.Weassumethatthereceiverhasthesameamountofresourcesavailableasthesender.Letdenotetheresourcesallocatedtoalternativeconditionalonobtainingsignal,withthebalance1allocatedtoThesenderÕspayoffisaweightedaverageofhisinvestmentsinthetwoalternatives:q,yyx(p(x)y+(1p(x))y)]+(1q)[1x(p(x)(1y)+(1p(x))(1y))]=(2q1)[x(p(x)y+(1p(x)y)],where[0,1].HereindexesthesenderÕstypeandmeasureshispreferencesforalternativeoveralternative.Intheabsenceofareceiver,hightypesobtainahigherpayofffrominvestinginalternativeandlowtypesobtainahigherpayofffrominvestinginalternative.Thecumulativedistributionofisdenotedby.Itisassumedtobeatomless,withsupport[0,1],andadensityfunctionwhichissymmetricaround .Inthepresenceofareceiver,thesenderÕsproÞtsfromeachalternativeincreaseswiththesizeofhisowninvestmentanddecreaseswiththesize4.Condition(iii)isnotessential.Ifthesignalisrandomevenwithoutanexpenditureofresources,thesenderreceivessomefreefeinting.Thisrandomnessmaybeenoughforthesender,inwhichcasenofeintingoccurs,butotherwisetheanalysisisverysimilartothepresentresults.Thisissuewasexploredinanearlierversionofthepaper. behaviorandpayoffsvarywiththesefactors.ProofsofthepropositionscanbefoundintheAppendix.4.1TheValueofRiskAversionIsarisk-aversereceiveraweakeropponentthanarisk-neutralreceiver?Theanswerdiffersdependinguponwhetherthesignalingtechnologyisrevealingornoisy.Intherevealingcase,thesenderÕspayoffis 21x1 Thus,hispayoffdependsuponbutdoesnotdependonthereceiverÕspreferences.ThereasonisthatthereceiverÕsstrategyisdeterminedbytheconditionthatthesendermustbeindifferentbetweenfullinvestmentandthepartialinvestmentofThereceiverÕsstrategydoesdependuponhispreferenceswhenthetechnologyisnoisy.Considertworeceivers,1and2,withutilityandassumethatreceiver1ismoreriskaversethanreceiver2.ApplyingTheorem1ofPratt(1964),forany 2,U1(y) U1(1y)U2(y) Thus,givenanypurestrategybythesender,themorerisk-aversereceiveralwaysdivideshisresourcesmoreevenlybetweenthetwoalternatives,whichyieldsahigherpayofftothesender.Inequilibrium,thesendertakesadvantageofthereceiverÕsriskaversionbyinvestingmoreinhispreferredalternativeandearningahigherpayoff.Proposition3:Ifthetechnologyisnoisy,thesenderpreferstoplayagainstamorerisk-aversereceiver.Ifthetechnologyisrevealing,thesenderÕspayoffdoesnotdependuponthereceiverÕsdegreeofriskaversion4.2TheValueofStealthInourmodel,thesendertakesthesignalingtechnologyasexogenous.But,hemayhavesomescopeforchoosingamongsignalingtechnolo-gies.Forexample,inmodernwarfare,thesignalingtechnologydependsuponthemethodofattack.Adversariesarelikelytohaveahardertimedetectingthetargetofanairattackthananarmedforcesattack,andanairattackthatusesconventionalbombersismoreeasilydetectedthananairattackthatusesstealthbombers.Thequestionthatarisesis,inwhatcircumstances,ifany,doesthesenderpreferamorerevealing JournalofEconomics&ManagementStrategyhisresourcesandinvestinganamountthatislessthanhalfofhisresourcesinhispreferredalternative,andthereceiverrespondstothesignalgeneratedbyinvestingmorethanhalfofhisresourcesinalternativeagivensignalalternativebgivensignalWeinterprettheinvestmentbythesenderinthelesspreferredalternativeasafeint.Intheabsenceofsignals,theequilibriumwouldconsistofthesenderinvestingallofhisresourcesintothepreferredalternative,andtherisk-aversereceiverdividinghisresourcesequallybetweenthetwoalternatives.Withsignals,thesendercannotpursuehisdesiredobjectivewithoutrevealinghisplans.Asaresult,hetriestodisguisehistrueobjectivebyinvestingsomeofhisresourcesintothelesspreferredalternative.Thesolepurposeofthisinvestmentistomisleadthereceiverintoallocatinghisresourcesawayfromthemorepreferredalternative.Propostions1and2provideastrikingcharacterizationofequi-libriumbehavior.Whensignalsarenotveryinformative,thesenderalwaysfeints,thefeintisrelativelysmall,andissuccessfulonlysomeofthetime.ThesignalisinformativebecauseitchangesthereceiverÕsbeliefsandherespondsbyallocatingmoreresourcestothealternativesignaled.Whensignalsareinformative,thesenderonlyfeintssomeofthetime,butifhefeints,itisalwaysalargefeint,usingmorethanhalfofhisresources.Onecanshowthattheexpectedsizeoftheinvestment ,soitisrationalforthereceivertorespondtosignalinvestingmorethanhalfofhisresourcesinalternativeandtosignalbyinvestingmorethanhalfofhisresourcesinalternative.Asaresult,whetherthesenderinvestsalargeorasmallamountinhispreferredalternative,ittendstobematchedbythereceiver.Aninterestingspecialcaseariseswhenthereceiverisriskneutral.Inthiscase,thereceiveralwaysfullyinvestsinalterativeinguponwhetheröisgreaterthanorlessthan .Hence,tosupporttheparticulardivisionneededtomakethesenderindifferent,thereceiverhastobeindifferentoverallallocations.Heisindifferentifandonlyiftherandomizationonthepartofthesenderhastheeffectofmakingthesignalsuninformative(i.e.,ö ).Regardlessofwhichsignalisreceived,thereceiverÕsposteriorbeliefsarethesameashispriorbeliefs.Despitethisindifference,thereceiverhastoresponddifferentlytothesignalreceived.4.ComparativeStaticsThekeyprimitivesofourmodelarethesignalingtechnologyandthepreferencesofthereceiver.Inthissectionweexaminehowequilibrium JournalofEconomics&ManagementStrategyMilgrom,P.andJ.Roberts,1982,ÒLimitPricingandEntryUnderIncompleteInformation:AnEquilibriumAnalysis,Ó,50(2),443Ð460.ÑÑ,andÑÑ,1986,ÒPriceandAdvertisingSignalsofProductQuality,ÓJournalofPoli-ticalEconomy,94,796Ð821.Pratt,J.W.,1964,ÒRiskAversionintheSmallandintheLarge,Ó,32,122Ð136.Saloner,G.,1987,ÒPredation,Mergers,andIncompleteInformation,ÓTheRANDJournalofEconomics,18(2)(Summer),165Ð186.Sobel,J.,1985,ÒATheoryofCredibility,ÓReviewofEconomicStudies,LII(4),557Ð573.VonNeumann,J.andO.Morgenstern,1944,TheoryofGamesandEconomicBehaviorPrincetonUniversityPress.Yergin,D.,1991,ThePrize,Simon&Shuster,NewYork.YofÞe,D.B.andM.Kwak,2001,JudoStrategy,HavardBusinessSchoolPress. Giventheabovelemma,thesupportof)forlargerthan whereAnecessaryconditionfortobeabestreplyagainstastrategy()isthatitsatisÞesNotethatthesecond-orderconditionsfora(local)optimumaresatisÞedbecauseitislessthan .Furthermore,because)for,thesenderÕspayofffrom1isthesameashispayofffromthelatterisabestreply).Bysymmetryof,thesupportof)forlessthen 0,1where1)forgreater BydeÞnitionofamixedstrategy,thehightypesmustbeindifferentbetweenchoosingand1.Similarly,lowtypesmustbeindifferentbetweenchoosing0and1.Thesetwoindifferencerelationshipsyieldtwoequations,whichuniquelydeterminethereceiverÕsstrategy.Solvingtheequationsfor 2+1x1 TheÞnalstepoftheconstructionisforthesendertorandomizeinsuchawaythatthereceiverÕsbeliefsinduceshimtochooseconditionalonobtainingsignalconditionalonobtainingsignal.Letdenotetheprobabilitythathightypeschooseandassumethatthisisalsotheprobabilitythatlowtypeschoose1.TheÞrst-orderconditionfordeterminesthevalueof.Itsolves U1y=1öq() where 2+(1p(x1)Eq|q1 GivenAssumption2,thesolutionforisuniqueandliesbetween0and1.Furthermore,ö,whichimpliesthatthesenderÕsotherÞrst-orderconditionisalsosatisÞed.Wehaveestablishedthefollowingproposition.Proposition2:Supposethesignalingtechnologyisrevealing.Thereisauniqueequilibriuminwhichthesenderrandomizesbetweeninvestingallof6.Weassumethatisunique. wheretheinequalityfollowsfromDeÞnition1.Thisimpliesthat 1z�1pL2(z) ThedeÞnitionof ).Itthenfollowsfromequation(4)that.Q.E.D.ProofofProposition5.WithoutlossofgeneralitysupposethesenderÕstypeis2andletöandödenotethesenderÕsbestrepliestoastrategyinwhichthereceiverallocatesallofresourcestothealternativesignaledwhenthesignalingtechnologyisrespectively.Whenbothtechnologiesarenoisy,öandö andareequilibriumallocations.Therefore,lettingdenotethesenderÕsproÞtsfromtech-1)[ö1)[ö1)[öTheÞrstinequalityfollowsfromthedeÞnitionoföasabestresponseandthesecondfromthedeÞnitionoflessrevealing.Q.E.D.Banks,J.S.,1987,ÒAModelofElectoralCompetitionwithIncompleteInformation,ÓJournalofEconomicTheory,55,647Ð661.Bernheim,B.D.,1994,ÒATheoryofConformity,ÓJournalofPoliticalEconomy,102,841Ð927.ÑÑ,andS.Severinov,2003,ÒBequestsasSignals:AnExplanationfortheEqualDivisionJournalofPoliticalEconomy,111(4),733Ð764.Crawford,V.,2003,ÒLyingforStrategicAdvantage:RationalandBoundedlyRationalMisrepresentationofIntentions,ÓAmericanEconomicReview,93,133Ð149.ÑÑ,andJ.Sobel,1982,ÒStrategicInformationTransmission,Ó,50,1431ÐFarrell,J.,1993,ÒMeaningandCredibilityinCheap-TalkGames,ÓGamesandEconomic,5,514Ð531.Fudenberg,D.andJ.Tirole,1986,ÒAÔSignal-JammingÕTheoryofPredation,ÓRandJournalofEconomics,17(2),366Ð376.HatÞeld,T.,1997,ÒTheCrucialDeception,ÓUniversityofTexasDiscoveryMagazine,14(2),17Ð25.Langinier,C.,2000,ÒInnovation,ImprovementandStrategicPatentingDecision,ÓMimeo.McAfee,R.P.,2002,CompetitiveSolutions:TheStrategistÕsToolkit,PrincetonUniversityPress.Matthews,S.A.andL.Mirman,1983,ÒEquilibriumLimitPricing:TheEffectsofPrivateInformationandStochasticDemand,Ó,51(4),981Ð996. JournalofEconomics&ManagementStrategyProposition1:Thereisanequilibriuminpurestrategiesifandonlyifthesignalingtechnologyisnoisy.Theequilibriumisuniqueandconsistsofthesenderallocatingmoreresourcestohispreferredalternative,andthereceiverallocatingmoreresourcestoalternativeaorbdependinguponwhetherheobtainssignalToestablishnecessity,supposethesignalingtechnologyisreveal-ing.Assumption2impliesthatthebestreplyofthesendertoastrategyinwhichistoinvestlessthanhalfofhisresourcesinhispreferredalternative.Thesenderdoessoinhopesthathisactionwillinducethereceivertoinvestmostofhisresourcesintheotheralternative.Butthereceiver,knowingthesenderÕsstrategy,understandsthatthesignalhehasreceivedismorelikelytohavebeengeneratedbythefeint,indicatingthatthepreferredalternativeistheoppositeoftheoneimpliedbythesignal.Hence,heisnotfooled,andallocatesmoreofhisresourcestoalterativeconditionalonobtainingsignal,andtoconditionalonobtainingsignal.But,giventhisresponse,thesenderÕsoptimalityconditionsarenotsatisÞed.Thefailureofapurestrategyequilibriumtoexistinthecaseofrevealingtechnologiesmeansthatweneedtosearchforanequilibriuminmixedstrategies.Wewilllookforanequilibriuminwhichthesenderrandomizesbetweeninvestingasmallandalargeamountinhispreferredalternative.Thisrandomizationwillcausethereceivertobeuncertainastowhetherthesenderisahighorlowtype,andhenceaboutwhichofthetwoalternativesishispreferredone.ThemaindifÞcultyinconstructingamixedstrategyequilibriumwithacontinuumoftypesisthateachsendertypehastobeindifferentbetweeninvestingasmalloralargeamountofresourcesinhispreferredalternative.ThenextlemmaestablishesthatwecanrestrictoursearchformixedstrategiestopointsonthelowerandupperboundoftheconvexhullofthegraphofLemma3:Ifq ,Epifq ,EpAformalproofisgivenintheAppendix.Theideaissimple.Forgreaterthan ,consideranyallocationstrategythatgeneratesapointthatisnoton.BydeÞnitionof,thereisanalternativestrategysuchthat)andItthenfollowsfromthelinearityofthesenderÕsproÞtfunctionthatisamoreproÞtablestrategythan.HencethereexistsatleastonepurestrategyinthesupportofthatyieldshigherproÞtsthan.Theargumentfor issimilar. JournalofEconomics&ManagementStrategy (1öx)1 2+p)1 Applyingcondition(ii)ofAssumption1,weneedtoshowthat Exploitingthesymmetryof ,theaboveinequalityisequivalenttoshowingthat whichholdssinceö .Ontheotherhand,ifthetechnologyisrevealing,then (1öx)1 2+p)1 Inthiscase,anargumentsimilartotheonegivenaboveleadstoaProofofLemma3.Consideranystrategythatgeneratesapointthatisnoton.BydeÞnitionof,thereisanalternativestratgeysuchthat)andandEp(öXa)y+(1Ep(öXa))y]�EXa[Ep(Xa)y(1Ep(Xa))y]=Ea(Xa,y,y).ButthisimpliesthatthereisatleastonepurestrategyinthesupportofwhichyieldshigherproÞtsthan,whichcontradictsthehypothesisisabestreplyto().Theargumentfor(ii)issimilar.ProofofProposition4.Whentechnologyisrevealing,theequilibriumpayofftothesenderis1.Therefore,weneedtoshowthattheequilibriumresponsebythereceiverissmalleragainstamorerevealingtechnology.Letdenotethelowerboundoftheconvexhullofthegraphofandletdenotethesolutionto).Then,forany(1/2,1], JournalofEconomics&ManagementStrategyLetö)denotetheexpectationofconditionalonsignalandstrategy.Then,givensignalandstrategy,thereceiverÕsexpectedpayofffromallocatingtoalternativeABayesianNashequilibriumisaproÞlethatsatisÞestheusualconditions.ThereceiverÕsresponsemustbeoptimalgivenhisbeliefsandhisbeliefsmustbeBayesconsistentwiththesignalingtechnologyandthesenderÕsequilibriumstrategy.Thus,)for.ThesenderÕsallocationmustbeanoptimalresponseto(),whichimpliesthateverystrategyinthesupportofIntheabovemodel,thesenderÕstypeisexogenous.Inanearlierversionofthispaper,weendogenizedthesenderÕstypebyallowinghimtoÞrstchoosetheweight,restrictingittobe0or1,andthentochoosehisallocationofresources.Theendogenoustypemodeliswellsuitedtomilitaryexamples,aswediscussinalatersection.Theanalysisisquitesimilar,primarilybecause,inequilibrium,thesenderrandomizesbetweenthetwoalternatives,whichinturnleadstoasubformorsubgameequivalenttothemodelwithexogenoustypes.Consequently,theresultsarealsosimilar.3.TheEquilibriumWebeginouranalysisbydistinguishingbetweentwokindsofsignalingtechnologies.Figure1illustratesasignalingtechnologythatisnotveryresponsivetothedifferentialininvestment.Thethindashedlineisthediagonalandtheboldlineisthegraphof).Formostallocations,bothsignalsarenearlyequallylikely.Hence,aslongasthesenderallocatessomeresourcestobothalternatives,thesignalisnotveryinformative.Figure2illustratesasignalingtechnologythatisverysensitivetoanydifferencesintheamountsinvested.Thereceiverisalmostcertaintogetsignal andsignalislessthan .Thus,formostallocations,thereceivercanidentifythealternativeinwhichthesenderinvestsrelativelymoreresources.Theabovedescriptionsuggeststhefollowingcriterionforcompar-ingdifferentsignalingtechnologies.Letdenotetwosignalingtechnologieswheretheprobabilityofgeneratingsignalishigherthanthatofforanythatexceeds .Bysymmetry,theprobabilityofgeneratingsignalislessthan isalsohigherthanunder.Inthiscase,wewillsaythatisamorerevealingsignalingtechnologythan DepartmentofEconomicsUniversityofTexasAustin,TX78712R.PRESTONHumanities&SocialSciencesCaliforniaInstituteofTechnologyPasadena,CA91125Inbotheconomicandmilitarysituations,agentsmaytrytomisleadrivalsabouttheirtruetypesorplans,whatevertheymaybe.Weconsiderasimplemodelinwhichoneplayerattacksandtheotherplayerdefends.Weshowthatsuchenvironmentshavetwotypesofpossibleequilibriumbehavior,dependinguponthesignalingtechnology.IfthesignalisnotveryrevealingabouttheattackerÕsplans,thentheattackeralwaysinvestsmoreresourcesinattackthaninmisdirection.Ifthetechnologyisrevealing,thentheattackerdoesnotalwaysfeint,butwhenhefeints,heinvestsmorethanhalfofhisresourcesintomisdirection.Comparativestaticsalsodependonwhetherthetechnologyisrevealing.ÒAlwaysmystify,misleadandsurprisetheenemy,ifpossible.ÓÑGeneralThomasJ.ÒStonewallÓJacksonÒCreatehavocintheeastandstrikeinthewest.ÓÑSunTze1.IntroductionInthepasttwodecades,economistshaveusedsignalingmodelstorationalizeandunderstandanumberoffrequentlypractisedbusinessstrategies.Forexample,MilgromandRoberts(1982)showthatlimitpricingcanbeexplainedasanattemptbyanincumbentÞrmtosignalWethankaudiencesatNorthwestern,Arhus,Heidelberg,Rochester,Wisconsin,ITAM,UC-SanDiegoand,inparticular,HughNeary,EricMaskin,LarrySamuelson,GuofuTan,anonymousreferees,andanassociateeditorforhelpfuldiscussionsandcomments.KenHendricksthanksSSHRCCforÞnancialsupport.2006,TheAuthor(s)JournalCompilation2006BlackwellPublishingJournalofEconomics&ManagementStrategy,Volume15,Number2,Summer2006,431Ð456 itmorelikelythattheentrantwillstayout.Bernheim(1994)andBanks(1987)studyenvironmentsinwhichallsendertypeswishtobeperceivedasthemiddletype.InCrawfordandSobel(1982),eachsendertypewantstobeperceivedasahighertype,butnotnecessarilythehighesttype.BernheimandSeverinov(2003)studyamodelofbequestsinwhichthemostdesiredtypemaybehigherorlowerthanthesenderÕstype,butthedesiredtypeisstillanincreasingfunctionofthesenderÕstruetype.Modelsinwhichthedesiredtypeismonotoneincreasinginactualtypetypicallypossessaseparatingequilibriuminpurestrategiesinwhichessentiallyallsendertypesinvestinthesignal.Bycontrast,inoursignalingmodel,thedesiredtypeisadecreasingfunctionofthesenderÕstruetype:ÒhighÓtypeswanttobeperceivedasÒlowÓtypesandÒlowÓtypeswanttobeperceivedasÒhighÓtypes.Wearenotawareofanyonewhohasstudiedthisclassofsignalinggames,withorwithoutnoisysignals.Thepaperisorganizedasfollows.InSection2,wedevelopasimplemodeloffeints.InSection3,wecharacterizetheequilibrium.InSection4weprovidesomecomparativestaticresults.WediscussapplicationsinSection5andconcludeinSection6.2.TheSignalingModelThegenericsignalingmodelconsistsoftwoplayers,onesenderandonereceiver,inwhichthesenderÕstypeisprivateinformation.ThesenderactsÞrst,andthereceiveractssecondafterobservinganoisysignalabouttheactiontakenbythesender.ThepayoffstoeachplayerdependuponthesenderÕstypeandtheactionstakenbythetwoplayers.Inourmodel,eachplayerÕsactionconsistsofadivisionofresourcesbetweentwoinvestmentalternatives,.Letdenotetheresourcesthatthesenderallocatestoalternative.WenormalizethesenderÕstotalavailableresourcestoone,sothatthebalance1isallocatedto.Thereceiverthenreceivesaprivate,binarysignalabouttheallocationofthesenderÕsresources.Thesignalyieldsinformationabouttherelativeallocationofresources,thatis,whichalternativeisgettingmoreresources.DeÞne2.MatthewsandMirman(1983)introducednoisysignalsintothestandardsignaling3.VonNeumannandMorgenstern(1944)introducedthetermÒinvertedsignallingÓtodescribeenvironmentsinwhicheachsendertypewantstosignaltothereceiverthatitisthetype.Theirmainexampleispoker,whereaweakplayerwantstogivea(false)impressionofstrengthandastrongplayerwantstogivea(false)impressionofweakness.Ofcourse,theyconsideronlyzero-sum,completeinformationgames. Feints439 FIGURE1.THENOISYTECHNOLOGYCASEDeÞnition1:ismorerevealingthanp 2|�|p2(x)1 Inwhatfollows,wewilldistinguishbetweentwotypesofsignal-ingtechnologies:noisyandrevealing.WedeÞnethemrelativetotheidentitysignalingtechnology,DeÞnition2:SupposepThenpisanoisysignalingtechnologyifitislessrevealingthanpanditisarevealingsignalingtechnologyifitismorerevealingthanpInanoisytechnology,)exceedsforanylessthan and,inarevealingtechnology,)islessthanforanylessthan .Symmetryimpliesthattheoppositeinequalitiesholdforany and1.AnoisysignalingtechnologyisillustratedinFigure1andarevealingtechnologyisillustratedinFigure2. JournalofEconomics&ManagementStrategyunderstandsthatthesignalislikelytobegeneratedbythefeintandinfersthatthepreferredalternativeistheoppositeoftheoneindicatedbythesignal.Herespondsaccordingly,whichmakeslargefeintsapoorstrategy.Theequilibriumstrategyforsendersistorandomize,sometimesinvestingallofhisresourcesandothertimesinvestinglessthanone-halfofhisresources,inthepreferredalternative.TherandomizationensuresthathisadversaryisuncertainaboutthesenderÕstype,andaboutthescaleofthesenderÕsinvestmentconditionalonhistype.Thus,inourmodel,allsendertypesfeint,andthefeintsaresometimesquitesuccessful.Ourmodelcanexplainbothfrequent,smallfeintsandinfrequent,largefeints.Thecomparativestaticresultsforthetwocasesarestrikinglydifferent.TheriskaversionofthesenderÕsrivalworkstothesenderÕsadvantagewhenthesignalisnoisybutisirrelevantwhenthesignalisrevealing.Itmightseemthatamorerevealingsignalwouldalwayshurtthesenderbecausethesenderisbestoffwhenhecanactsecretlywithoutgeneratinganysignal.Thisintuitionisbasicallycorrectfornoisysignalingtechnologies.But,whenthetechnologyisrevealing,theinformativenessofthesignalworkstothesenderÕsadvantage.Thereasonisthatamorerevealingsignalmakesiteasierforthesendertofeint,andhencemoreproÞtable.CrawfordandSobel(1982)andSobel(1985)showthatnocommu-nicationoccursinsender-receivergameswheremessagesarecostless(ÒcheaptalkÓ)andthereceiverÕsgainisthesenderÕsloss.Farrell(1993)usedthetermÒbabblingÓtodescribeequilibriaofcheaptalkgamesinwhichthesenderÕsmessageisuninformative.Crawford(2003),inapapermotivatedbyouranalysis,introducesboundedlyrationaltypesintoazero-sum,cheaptalkgameandshowsthattheirpresenceyieldsequilibriainwhichthesophisticated(fullyrational)senderliesandfoolsasophisticated(fullyrational)receiversomeofthetime.ÒBabblingÓequilibriadonotariseinourmodelbecausesignalingiscostly.Ifthereceiverignoresthesignal,thesenderÕsbestreplyistoapplymoreresourcestothepreferredalternative,becauseresourcesdevotedtofoolingthereceiveraremostlywasted.But,inthatcase,thereceivershouldnotignorethesignalbutallocatemoreresourcesatthealternativesignaled.Thus,intheequilibriumofourmodel,thesignalisinformative,inthatthereceiverÕsBayesupdatediffersfromhispriorandheallocatesmoreresourcestothealternativeindicatedbythesignal.Ourmodelandresultscontributetosignalingtheory.Thestandardassumptioninsignalingmodelsinindustrialorganizationisthatallsendertypeswishtobeperceivedasthesametype.Forexample,inMilgromandRobertsÕ(1982)modelofentryandinFudenbergandTiroleÕs(1986)modelofsignaljamming,allincumbentÞrmtypeswanttobeperceivedashavingthelowestcostbecausethisinferencemakes OperationFortitude.OperationFortitudewasafeint.Aspartofthisoperation,amythicalÒU.S.1stArmyGroup,ÓledbyGeorgePatton,wascreated.ThisarmyconsistedofplywoodairplanesandinßatabletankslocatednearDover,andavastarmadaofrubberlandingcraftlocatedintheThamesRiverestuary.PattonÕsmythicalforceproducedlargevolumesofcodedradiotrafÞcforGermaninterception,andtherewereextensiverealmilitarymaneuversnearthelocationoftheersatzarmy.Falseinformationwasfedtoknownenemyagents.Theoperationworked.TheGermanhighcommand,ledbyFieldMarshallWilhelmKeitel,believedthattheinvasionofNormandywasafeint,adiversionfromtherealinvasiontocomeatCalais.NineteenGermandivisionsremainedatPasdeCalaisafterD-Day,theinitiationoftheNormandyinvasion.TheAlliescontinuedbombingCalaisevenaftertheinvasionbegan,withtwobombsdroppedatCalaisforeverybombdroppedatNormandy,successfullymaintainingtheruseforseveralcriticaldays.Wemodelfeintsasasender-receivergamewithnoisysignals.Thesenderhastoallocateresourcesbetweenatleasttwoalternatives.ThesenderÕstypedetermineswhichalternativeispreferred.ThereceivercannotobservethesenderÕstypebuthedoesobserveanoisysignalofthesenderÕsallocationandcandrawinferencesabouthistypefromtheobservedsignal.ThereceiverÕsinferencematterstothesenderbecausethereceiverÕsgainishisloss.Thus,eachsendertypehasanincentivetomisleadthereceiver.Thedistinguishingfeatureofourmodelisthatthewayinwhicheachtypetriestomisleadthereceiveristopretendtobehavelikethetype.Forexample,theAllieswantedtheGermanstothinkthattheywereattackingPasdeCalaisbut,iftheyhadattackedPasdeCalais,theywouldhavewantedtobeperceivedasattackingNormandy.Similarly,Tidewaterwantedtobeperceivedasbuildingthepipelineontheroutecoveredbythefakesurveys,buthaditintendedtobuildthepipelinealongthisroute,itwouldhavewantedtobeperceivedasbuildingthepipelineontheroutethatitactuallychose.Asinmostsignalingmodels,weareinterestedinknowingwhenandhowmuchÞrmsarelikelytoinvestinsignaldistortion,andwhetherrivalsaremisledintotakingactionsthatfavorthesender.Ourmajorresultdistinguishestwocases.IfthesignalgeneratedbythesenderÕsactionisnotveryÒrevealingÓ(inthesensetobemadeprecise),thenthesenderpursuesapurestrategyandallocatesmostofhisresourcestothepreferredalternative.Feintsaresmallanddeterministic.Incontrast,whenthesignalisveryrevealing,thesendercannoteasilyconcealhisactionsandsmallfeintsarenoteffective.Moreover,large,deterministicfeintsarealsonoteffective.Thereceiver,knowingthesenderÕsstrategy,1.Formoredetails,seeHatÞeld(1997). JournalofEconomics&ManagementStrategythatitscostsarelowandhence,thatentryislikelytobeunproÞtable.Theirmodelcan,withsmallchanges,bereinterpretedtoexplainpreda-torypricing.Inamodel,whereentrantsusetheircurrentproÞtstoinferfutureproÞts,FudenbergandTirole(1986)showthattheincumbentÞrmwilltrytomanipulatetheentrantÕsinferencebyloweringitsprice,therebyencouragingtheentranttoexitthemarket.Saloner(1987)usesavariantoftheMilgrom-RobertsmodeltoexplainwhypredatorypricingcanloweraÞrmÕscostsofacquiringarival.FollowingtheseminalworkofMilgromandRoberts(1986),alargeliteraturehasdevelopedshowinghowÒburningmoneyÓintheformofimageadvertizingcansignalconsumersthatthefeaturedproductishighquality.Thispaperdevelopsanewkindofsignalingmodeltostudyfeints,acommonlyusedstrategyinconßicts.Feintsareoffensiveactionscon-ductedwiththepurposeofdeceivingrivalsastothelocation(ortime)ofthemainoffensiveaction.Theyareubiquitousinmilitarycampaignsandsports,butalsooccurfrequentlyinpoliticalandlegalconßicts.Inbusiness,feintsarelikelytooccurwhenaÞrmistryingtoenteranewmarketormarketanewproductbutwishestoconcealthetruenatureofitsinvestmentplanstopreventarivalfromcounterattackingeffectivelywhileitisstillvulnerable.Forexample,bytheendofthe1870s,theStandardOilTrusthadlockedupmostofthetransportationandreÞningofoilinPennsylvania,atthattimetheworldÕslargestproducingregion.AgroupofproducersattemptedtocircumnavigateStandardOilbybuildinga110-milepipelinetoconnecttheoilregionstothePennsylvaniaandReadingRailroad.Thisplanwasaudacious,becausenoonehadeverattemptedsuchalargepipeline,butalsobecausethegroupattemptedtokeepStandardOilinthedarkaboutthescaleandlocationofthepipeline.FakesurveysweretakeninordertoconfuseStandardOiloftherouteofthepipelinesothatitcouldnotblockit.InMay1879,theTidewaterpipelineopened.ItÒnotonlycaughtStandardbysurprise,butalsomeantthatitscontroloftheindustrywassuddenlyagaininjeopardyÓ(Yergin,1991,p.43).StandardacquiredTidewaterin1883.OtherexamplesincludepharmaceuticalÞrmspatent-ingÒdead-endÓproducts(Langinier,2001)tomisleadcompetitorsÕR&DexpendituresandoilÞrmssurveyingtractsthattheyhavenointentionofbiddinginordertoconfusepotentialrivalsastowhichareastheyshouldpursue.AfamousmilitaryexamplewastheAlliedinvasionofNazi-occupiedFrance.ThemostnaturallocationsfortheinvasionwerePasdeCalaisandNormandy.TheofÞceoftheChiefofStaffoftheSupremeAlliedCommander(COSSAC)decidedonNormandy.However,inordertomisleadtheNazimilitaryintocontinuingtofortifyCalaisattheexpenseofNormandy,theAlliesdevelopedanambitiousplannamed Lemma1:Tosolvefortheequilibriumofthesignalinggame,weproceedbybackwardinduction.GiventhesenderÕsstrategy,thereceiverÕsbestreplyiseasilycharacterized.DifferentiatingwithrespecttoÞrst-orderconditions, U(1ys)=1öqs ManipulatingtheseÞrst-orderconditionsyieldsthefollowingresult.Lemma2:ifandonlyifConsidernextthesenderÕsbestreplytoanarbitraryreceiverstrategy.If,thenthesenderÕspayoffisstrictlydecreasingin ,andstrictlyincreasingin .Thus,thesenderÕsbestreplyinthiscaseistoallocateallofhisresourcestothealternativethathasthehigherweightinhisutilityfunction.Butthisimpliesthatö,whichcontradictsthepreviouslemma.Hence,anecessaryconditionforexistenceofapurestrategyequilibriumisthatGivenanysuchstrategy,andassuminganinteriorsolution,thesenderÕsbestreplymustsatisfy Thereare(generically)twopossiblesolutionstotheÞrst-ordercondi-tion:oneliesinthehalfinterval[0 ]andtheotherliesinthehalfinterval 1].Letö)denotethesolutionthatexceeds .Symmetryimpliesthattheothersolutionis1greaterthan thesecond-orderconditionholdsatö)ifisnoisyandat1)ifisrevealing;theconverseistrueforlessthan Inthecaseofnoisytechnologies,thebestreplyforasenderistoallocatemorethanhalfofhisresourcestoalternative andtoalternativeislessthan Becausehightypesaremorelikelytogeneratesignal,andthepriorexpectationis ,thereceiverbelievesthatheismorelikelytobeplayingagainsthightypesconditionalon.Similarly,becauselowtypesaremorelikelytogeneratesignal,thereceiverÕsposteriorbeliefsconditionalonsignalisthatheismorelikelytobeplayinglowtypes.Giventhesebeliefs,itfollowsfromthereceiverÕsÞrst-orderconditionsthatheallocatesmorethanhalfofhisresourcestoalternativeifheobtainssignalandtoalternativeheobtainssignal,satisfyingthenecessaryconditionforexistence. JournalofEconomics&ManagementStrategy FIGURE2.THEREVEALINGTECHNOLOGYCASEByrandomizingtheallocationofresources,thesendercanachievesignalingprobabilitiesthatdifferfrom.Thesetofprobabilitiesthatcanbegeneratedusingmixedstrategiesistheconvexhullofthegraphof.DeÞneas,respectively,thelowerandupperboundsoftheconvexhullofthegraphof.TheseboundsaredepictedasthethickdashedlinesinFigures1and2.Notethat,byAssumptions1(ii)and(iii),theconvexhullincludesthediagonal.Thefollowinglemma,whichfollowsfromAssumption1(ii),establishesthattheboundariesaresymmetric. notwhethertoenterthemarketbutwhichmarkettoenter,thentheentrygamebecomesafeintinggame.Entryinvolvesasubstantialinvestmentinmarketing,andthepayoffstothepotentialentrantareaweightedaverageofthedifferencebetweenitsmarketingexpendituresandthoseoftheincumbentineachmarket.ThemarketweightrepresentstheentrantÕstype,andisameasureofitscomparativeadvantageinenteringaparticularmarket.IftheentrantignorestheresponseoftheincumbentÞrm,thentheentrantÕsbeststrategyistouseallofitsscarceresourcestoenterthemarketinwhichithasacomparativeadvantage.However,itknowsthatpursuingsuchastrategywouldgenerateaquickandoverwhelmingresponsefromtheincumbent.AbetterstrategywouldbetoexploittheincumbentÞrmÕsuncertaintyaboutwhichmarketistherealtargetbyinvestinginbothmarkets.Thiswillforcetheincumbenttosplititsresourcesbetweenthetwomarkets,andweakenitsdefenseofthetargetmarket.Hopefully,theincumbentwillallocaterelativelymoreresourcestodefendingtheothermarket.Whethertheentranthastoexpendasmalloralargeamountondiversionarymarketingwilldependupontheobservabilityofitsexpenditures.Theimportantissueisthat,eveniftheincumbentÞrmcaneasilyidentifythemarketinwhichtheentrantislaunchingabiggercampaign,theequilibriumdoesnotinvolvehead-to-headcompetition.Thebiggermarketingcampaignissometimesadiversion,andknowledgethatthismightbethecaseforcestheincumbentÞrmtoexpendresourcesinbothAvariantoftheabovemodelthatwestudiedinapreviousversionofthispaperinvolvesnewproductentryovertime.SupposeaninnovativeÞrmhasperfectedanewproductanditsrival,knowingthattheentranthasbeenworkingonanewproduct,ispositionedtoproduceaclosesubstituteassoonastheentrantÕsnewproducthitsthemarket.Insteadofengaginginahead-to-headcompetitionwithitsrival,theinnovatorcouldpursueamoresubtlestrategy.ItcouldmarketaninferiorversionofitsnewproductÞrst,inthehopethatitsrivalwillexpendallofitsscarceresourcesondevelopingandmarketingaclosesubstitutefortheinferiorproduct.Oncetherivaldoesso,thentheentrantcomesoutwiththeoriginalproduct,callingitthenew,improvedversion,andcapturesthemarket.Ofcourse,inequilibrium,theimitatorÞrmanticipatestheinnovativeÞrmÕsstrategy,anddoesnotexpendallofitsresourcesreactingtotheÞrstproductmarketedbytheentrant.ItkeepssomeresourcesinreserveonthechancethattheÞrstproductwasadecoyandithastoreacttothenew,improvedversion.Nevertheless,bypursuingsuchastrategy,theinnovatorforcesitsrivaltodelayorreduceitsresponsetotheinitialproductoffering,whichincreasesproÞtsevenwhenitisnotadecoyproduct.ThefakesurveysconductedbyTidewater JournalofEconomics&ManagementStrategytheattackerfromasneakattackissimilartohispayofffromanall-out5.2EntryGamesInthestandardsignalingmodelofentry,thesenderisanincumbentÞrmandthereceiverisapotentialentrantwhohastodecidewhetherornottoenterthemarketandcompeteagainsttheincumbentÞrm.Itiswillingtoenteragainsthigh-costincumbentsbutnotagainstlow-costincumbents.Thus,thehigh-costincumbenthasanincentivetoimitatethelow-costincumbentÕsbehaviorandtrytomisleadtheentrantintostayingoutofthemarket.However,inmanycases,therolesofthesenderandreceiverarereversed:itistheentrantwhowantstomisleadtheincumbentÞrm.Forexample,CapitalOne,aninnovativecreditcardissuer,wantedtooffernewproductstargetedatspeciÞccustomersegments.Inordertodoso,theycollecteddetailedinformationonitscustomersandthentestedthousandsofofferingstoidentifythenewmarkets.ThesuccessofthisstrategydependeduponthecompanyÕsabilitytokeeptheindustrygiantssuchasCitibankfrommatchingitsproductofferings.WaryofdrawingCitibankÕsattention,CapitalOnepursuedamorecostlymarketingstrategythatreliedupontelephoneso-licitationanddirectmailratherthanhigh-proÞleadvertisingcampaigns.AccordingtoaformerexecutiveofCapitalOne,competitorsmaycomeacrossitsmailingsbutÒCitibankwouldneverknowwhatcustomerswearetargeting,unlesswetoldthem.Asaresult,wehavelargelybeenabletostayundertheradarscreen.ÓPalmpursuedasomewhatsimilarstrategyinmarketingthePilot.Thecompanyintroduceditsproductin1996,carefullypositioningitasacomplementtothePC,notasubstitute,andplayingdownitspotentialinordernottodrawunwelcomeattentionfromcompetitorslikeMicrosoft.Thestrategyapparentlyworked;MicrosoftunderestimatedtheproÞtabilityofthehandheldmarketanddelayeditsattackonPalmuntilitwastoolate.TheCapitalOneandPalmstoriesareexamplesofinnovativeÞrmsundermarketingnewproductstodisguisetheirentryintoanincumbentÕsmarkets.Buttheyarenotexamplesoffeintsbecause,iftheÞrmshadnotbeeninnovative,theywouldhavehadnoreasontopretendthattheywereinnovative.But,ifthedecisionbytheentrantis8.AclassicexampleinsportsofsuchastrategyistherunningplayinAmericanfootballknownastheÒnakedquarterbackbootleg.ÓWhiletheentireoffensivelineandrunningbacksmoveinonedirection,thequarterbackfakesahandofftoarunningback,tuckstheballbehindhisleg,andjogsoffintheoppositedirection.Hehasnoblockerstohelphimgainanyyardsbut,ifthefeintissuccessful,noneisrequired.9.ThetwocasesarestudiedindetailinYofÞeandKwak(2001).WethankLucyWhitefordrawingourattentiontothisbook.